Analysis of coolant entrance boundary shape of porous region to control cooling along exit boundary

A cooled porous region has a plane surface exposed to a specified spatially varying heat flux. The coolant leaves the region through this surface, and it is desired to control the flow distribution to maintain a specified uniform surface temperature. This is accomplished by having the coolant entrance surface shaped to provide in the region the necessary variation of path length and, hence, flow resistance. The surface shape at the coolant entrance is found by solving a Cauchy boundary value problem. An exact solution is obtained that will deal with a wide variety of heating distributions for both two- and three-dimensional shapes.